On the Ising Model

In relation to Natalie Paquette’s essay, a more pedestrian example of physical mathematics may be of interest. André Weil once described a threefold analogy from Richard Dedekind and Heinrich Weber, comprised of arithmetic and combinatorics, algebra, and analysis. Dedekind and Weber sought to make Bernhard Riemann’s geometric and analysis account of algebraic functions more algebraic, and analogized the algebraic theory to arithmetic and the theory of numbers.

More recently, connecting harmonic analysis to number theory in the Langlands Program, what might be difficult to compute in one mode is straightforward in the other. Properties of the Riemann zeta function that package the prime numbers, an arithmetic object, are revealed by the elliptic theta function, an algebraic and analytic object—since, as Riemann showed, zeta and theta are linked by a Fourier-like Mellin transform.

In the last seventy-five years, physicists have exactly solved the two-dimensional Ising model of ferromagnetism in numerous ways, the solution being the partition function, whose logarithm is the chemist’s free energy, as well as other properties of the system, such as its spontaneous magnetization. The solutions may be combinatorial, algebraic, or analytic—depending on assumptions about smoothness and continuity. All the various solutions come to the same result. The two-dimensional Ising model is, in effect, the identity that allows for these three perspectives.

It would be interesting if there were a fruitful analogy between the threefold mathematical analogy and the threefold physical analogy. In each case, what is easy to compute by one method is difficult to compute by another.

Put differently, physicists have been exploring the mathematicians’ analogy for some time without being aware of it. They were calculating, often in ingenious ways. The mathematicians work hard to exploit their analogy, but the physicists’ effort is not much on their minds. The physicists have a simple model of the connections the mathematicians seek. The mathematicians make sense of the variety of ways of solving the Ising model, seeing them as part of a whole.

Martin Krieger

Martin Krieger is professor of planning at the Sol Price School of Public Policy at the University of Southern California.